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Glossary

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Transformations and SymmetryWallpaper Groups

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In the previous sections we have now seen two different kinds of symmetry, that correspond to two different transformations: rotations and reflections. But there is also a symmetry for the third kind of rigid transformation: translationsspinsflips.

Translational symmetry does not work for isolated objects like flowers or butterflies, but it does for regular patterns that extend into every direction:

Hexagonal honyecomb

Ceramic wall tiling

In addition to reflectional, rotational and translational symmetry, there even is a fourth kind: glide reflections. This is a combination of a reflection and a translation in the same direction as the axis of reflection.

A pattern can have more than one type of symmetry. And just like for squares, we can find the symmetry group of a pattern, which contains all its different symmetries.

These groups don’t tell you much about how the pattern looks like (e.g. its colours and shapes), just how it is repeated. Multiple different patterns can have the same symmetry group – as long are arranged and repeated in the same way.

These two patterns have the same symmetries, even though they look very different. But symmetries are not about colours, or superficial shapes.

These two patterns also have the same symmetries – even though they look more similar to the corresponding patterns on the left, than to each other.

It turns out that, while there are infinitely many possible patterns, they all have one of just 17 different symmetry groups. These are called the Wallpaper Groups. Every wallpaper group is defined by a combination of translations, rotations, reflections and glide reflections. Can you see the centers of rotation and the axes of reflection in these examples?